Actually, I just googled it and I did use wrong terminology.
It seems instead of "open-ended" or "close-ended", I am talking about a "lease-purchase" option, which it seems about 50/50 leases have - a right to exercise an agreed upon price for purchase.
It seems an open-ended lease revolves on what happens on the appraisal at the end. The leasee is bearing the risk. If it appraises lower, than you owe the dealership extra money. Theorectically, if it appraises higher, than they may owe you. (you can bet that hardly ever happens). In exchange for bearing the risk, you get a lower lease monthly payment.
Why am I considering it?
1. Want a mini-van with relatively low miles (in the 25-50K range, which is near the end of most leases)
2. I would be able to defer the "ballon payment" (the downpayment) until the end (with leases with an option to purchase) and thus earn money in an account in the meantime after I liquidated my car.
3. Tax-write off for business (a least partial)
4. I am a weirdo in that I am always looking for "creative solutions"
PS: I find these deals "enticing" in that you are getting away with never making a downpayment on the lease. Go to one of those websites though and see what's out there - really bad deals people made. And I'd say there are lots of Beamers too.